Here you will learn how to solve or evaluate limits at infinity with examples. Let’s begin – Limits at Infinity Algorithm to evaluate limits at infinity : 1). Write down the given expression in the form of a rational function i.e. \(f(x)\over g(x)\), if it is not so. 2). If k is the highest power […]
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Here you will learn what is the rationalisation method to solve or find limits with examples. Let’s begin – Rationalisation Method to Solve Limits This method is particularly used when either the numerator or denominator or both involve expression consisting of square roots and substituting the value of x the rational expression takes the form
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Here you will learn what is the factorisation method to solve limits with examples. Let’s begin – Factorisation Method to Solve Limits Consider the following limit : \(\displaystyle{\lim_{x \to a}}\) \(f(x)\over g(x)\) If by substituting x = a, \(f(x)\over g(x)\), reduces to the form \(0\over 0\), then (x – a) is a factor of f(x)
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Here you will learn direct substitution method to solve limits with examples. Let’s begin – Direct Substitution Method to Solve Limits Consider the following limits : (i) \(lim_{x \to a}\) f(x) (ii) \(lim_{x \to a} {\phi(x)\over \psi(x)}\) If f(a) and \(\phi(a)\over \psi(a)\) exist and are fixed real numbers, then we say that \(lim_{x \to a}\)
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Here you will learn some limits examples for better understanding of limit concepts. Example 1 : If \(\displaystyle{\lim_{x \to \infty}}\)(\({x^3+1\over x^2+1}-(ax+b)\)) = 2, then find the value of a and b. Solution : \(\displaystyle{\lim_{x \to \infty}}\)(\({x^3+1\over x^2+1}-(ax+b)\)) = 2 \(\implies\) \(\displaystyle{\lim_{x \to \infty}}\)\(x^3(1-a)-bx^2-ax+(1-b)\over x^2+1\) = 2 \(\implies\) \(\displaystyle{\lim_{x \to \infty}}\)\(x(1-a)-b-{a\over x}+{(1-b)\over x^2}\over 1+{1\over x^2}\) =
Here, you will learn what is squeeze theorem or sandwich theorem of limit and limit of exponential function with examples. Let’s begin – Squeeze Theorem (Sandwich Theorem) If f(x) \(\leq\) g(x) \(\leq\) h(x); \(\forall\) x in the neighbourhood at x = a and \(\displaystyle{\lim_{x \to a}}\) f(x) = l = \(\displaystyle{\lim_{x \to 1}}\) h(x) then
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Here, you will learn how to find limit of trigonometric functions and limits using series expansion with example. Let’s begin – Limit of Trigonometric Functions \(\displaystyle{\lim_{x \to 0}}\) \(sinx\over x\) = 1 = \(\displaystyle{\lim_{x \to 0}}\) \(tanx\over x\) = \(\displaystyle{\lim_{x \to 0}}\) \(tan^{-1}x\over x\) = \(\displaystyle{\lim_{x \to 0}}\) \(sin^{-1}x\over x\) [where x is measured in
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Here, you will learn how to solve indeterminate forms of limits and general methods to be used to evaluate limits with examples. Let’s begin – Indeterminate Forms of Limits \(0\over 0\), \(\infty \over \infty\), \(\infty – \infty\),\(0\times \infty\), \(1^{\infty}\), \(0^0\), \({\infty}^0\) Note : (i) Here 0, 1 are not exact, infact both are approaching to
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Here, you will learn definition of limit in calculus, left hand limit, right hand limit and fundamental theorem of limit. Let’s begin – Definition of Limit in Calculus Let f(x) be defined on an open interval about ‘a’ except possibly at ‘a’ itself. If f(x) gets arbitrarily close to L(a finite number) for all x
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